The solution types shown in Figure REF  have the general form that outside the DSW and the resonant wavetrain
(if they exist) the solution is non-diffractive and governed by the non-dispersive equations (REF ) and ().
There is an expansion fan which links the level \(u_{-}\)  to an intermediate level \(u_{i}\) , with the DSW (and resonant wavetrain)
then taking the solution from \(u_{i} = \sqrt{\rho _{i}}\)  to the level \(u_{+}\)  ahead. The expansion wave is given by the
simple wave solution on the characteristic \(C_{-}\)  of the Riemann invariant form (REF ) and () of the
non-dispersive equations with the Riemann invariant on \(C_{+}\)  constant. If we set the velocity of the trailing
edge of the DSW (or Whitham shock, depending on the regime) to be \(s_{i}\) , then this expansion wave solution is
[1]}, [2]}
\(|u| = \sqrt{\rho } = \left\lbrace  \begin{array}{cc}u_{-}, & \frac{x}{z} < -\frac{\sqrt{2}}{\sqrt{q}} u_{-}\\\frac{\sqrt{q}}{3\sqrt{2}} \left[ \frac{2\sqrt{2}}{\sqrt{q}}u_{-} - \frac{x}{z} \right], &-\frac{\sqrt{2}}{\sqrt{q}} u_{-}\le \frac{x}{z} \le \frac{\sqrt{2}}{\sqrt{q}} \left( 2u_{-} - 3\sqrt{\rho _{i}}\right) \\\sqrt{\rho _{i}}, & \frac{\sqrt{2}}{\sqrt{q}} \left( 2u_{-} - 3\sqrt{\rho _{i}} \right) < \frac{x}{z} \le s_{i} ,\end{array}\right.\) 
